The lattice geometry used in our simulation is depicted in Fig.REF a, where \(\hat{x}=(1,0)\)  and \(\hat{y}=(0,1)\)  denote two basis vectors of the square lattice. We consider cylindrical geometry with periodic (open) boundary condition in the \(\hat{y}\)  (\(\hat{x}\) ) direction in each layer. We focus on cylinders with width \(L_y=4\)  and length up to \(L_x=32\) , where \(L_y\)  and \(L_x\)  are number of sites along the \(\hat{y}\)  and \(\hat{x}\)  directions, respectively. The total number of sites of the system is \(2N\)  where \(N=L_x\times L_y\) . The hole doping concentration away from half-filling is defined as \(\delta =N_h/N\) , where \(N_h\)  is the number of doped holes and \(N_h=0\)  at half-filling. We keep up to \(m=25000\)  number of states in our simulation with a typical truncation error \(\epsilon \sim 10^{-6}\) . The on-site energy difference \(\varepsilon _0^{R\text{-Ni}}\)  plays a vital role in controlling the size of the \(R\)  electron pocket. Specifically, when \(\varepsilon _0^{R\text{-Ni}}=3.4\)  or \(3.7\) , the size of the rare-earth electron pocket matches well with the experimental results [1]}. We focus on the case with \(\varepsilon _0^{R\text{-Ni}}=3.4\)  in this study to make a direct connection with experiment. Similar results have been obtained for \(\varepsilon _0^{R\text{-Ni}}=3.7\)  and are provided in the supplementary material[2]}.
